# Approximation Results for Kinetic Variants of TSP

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## Abstract

We study the approximation complexity of certain kinetic variants of the Traveling Salesman Problem where we consider instances in which each point moves with a fixed constant speed in a fixed direction. We prove the following results.

- 1.
If the points all move with the same velocity, then there is a PTAS for the Kinetic TSP.

- 2.
The Kinetic TSP cannot be approximated better than by a factor of two by a polynomial time algorithm unless P=NP, even if there are only two moving points in the instance.

- 3.
The Kinetic TSP cannot be approximated better than by a factor of \( 2^{\Omega \left( {\sqrt n } \right)} \) by a polynomial time algorithm unless P=NP, even if the maximum velocity is bounded. The

*n*denotes the size of the input instance.

Especially the last result is surprising in the light of existing polynomial time approximation schemes for the static version of the problem.

## Keywords

Approximation Ratio Travel Salesman Problem Euclidean Plane Hamiltonian Path Kinetic Variant
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 1999